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<a NAME="E1:32_1_1"/><i><font color="Green">E69</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9">TOPREALB:9</a></i>;<br/>
<a NAME="E2:32_1_1"/><i><font color="Green">E70</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3">BROUWER:3</a></i>;<br/>
<a NAME="E3:32_1_1"/><i><font color="Green">E74</font></i>: 
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T17">TOPREAL9:17</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>;<br/>
<a NAME="E5:32_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:32_1_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/><a NAME="E4:32_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>;<br/><b>hence </b><a NAME="E5:32_1_1_1_1"/>
not <span class="p1">(<span class="default"><a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_1_1_1_2"/>
not <span class="p1">(<span class="default"><a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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